Center for Language Technology Njalsgade 80
DK-2300 Copenhagen anders@cst.dk
Abstract
Totally unordered or discontinuous com- position blows up chart size on most set- ups. This paper analyzes the effects of total unordering to type 2 grammars and simple attribute-value grammars (s- AVGs). In both cases, charts turn ex- ponential in size. It is shown that the k-ambiguity constraint turns charts poly- nomial, even for s-AVGs. Consequently, tractable parsing can be deviced.
1 Introduction
It is common knowledge among linguists that in many languages, the daughters of syntactic con- stituents can be locally reordered with little or no effect on grammaticality. Certain languages – of which Dyirbal and Warlpiri are often-cited mem- bers, but that also include Estonian and Finnish – exhibit a much more radical form of unordering, the kind of unordering that has made linguists propose
“crossed brances” analyses, e.g.
S
NP VP
Lisa John loved
yieldinglove(John,Lisa). The Finnish transla- tion is Liisaa Jussi rakasti. All six permutations of this sentence are grammatical.
Unordered grammars have been suggested in face of intra-constituent free word order. Similarly, a few
authors have proposed totally unordered grammars in face of free word order phenomena that involve discontinuous constituents. Dowty (1995), origi- nally published in 1989, is often cited as the original source.
This paper is structured as follows: Sect. 2 de- fines type 2 grammars and s-AVGs, and their totally unordered pendants. Sect. 3 establishes bounds on chart size for these kinds of grammars. Charts for totally unordered grammars are shown to be worst case exponential. In reply to this, Sect. 4 intro- duces thek-ambiguity constraint, which turns totally unordered charts polynomial again. Of course this means that polynomial time parsing can be deviced.
2 Grammars and total unordering
Our first task is to properly define the grammars in question:
2.1 Type 2 grammars
Definition 2.1 (Type 2 grammars). G = hN, T, P,{S}i is a type 2 grammar iff every production rule inP is of the form
A → ω whereA∈N andω∈ {N∪T}+.
Definition 2.2 (Derivability). For a type 2 grammar Gand ω1, ω2 ∈ (N ∪T)∗,ω1 =⇒1 ω2 iff there is a A → φ ∈ P and there are ψ1, ψ2 ∈ (N ∪T)∗ such that ω1 = ψ1Aψ2 and ω2 = ψ1φψ2. =∗⇒1
(the derivability relation) is the reflexive transitive closure of=⇒1.
Definition 2.3 (Type 2 languages). The language of a type 2 grammarGis defined as
L(G) = {x∈T∗ :S=∗⇒1 x}.
Definition 2.4 (Chomsky normal form). A type 2 grammarG=hN, T, P,{S}iis in Chomsky normal form iff each production has one of the following forms:
• A→BC
• A→a
• S →ǫ
wherea∈T andB, C ∈N − {S}.
Example 2.5. Consider the type 2 grammar with rules S → aXb|ab, X → aXb|ab. The Chomsky normal form of this grammar is obtained by adding the rulesA→a, B →band by reducing the length of theS, X-rules. Consequently,P′now includes:
S → AT|AB T → XB
X → AT|AB A → a
B → b
Lemma 2.6 (Equivalence of normal forms). Say G = hN, T, P,{S}i is a type 2 grammar. There is an algorithm to construct a grammar G′ = hN′, T, P′,{S′}i in Chomsky normal form that is weakly equivalent toG.
Proof. See Sudkamp (2005, 122–3).
2.2 Totally unordered type 2 grammars
Definition 2.7 (Totally unordered type 2 grammars).
G = hN, T, P,{S}i is a type 2 grammar iff every production rule inP is of the form
A → ω
whereA∈N andω∈ {N∪T}∗.
Definition 2.8 (Derivability). IfA=∗⇒1 ωandω′ ∈ permute(ω), thenA=∗⇒2 ω′.
Definition 2.9 (Totally unordered type 2 languages).
The language of a totally unordered type 2 grammar Gis defined as
L(G) = {x∈T∗ :S=∗⇒2 x}.
2.3 s-AVGs
s-AVGs are defined over simple attribute-value structures (s-AVSs):
Definition 2.10 (s-AVS). An s-AVS A is defined over a signature hAttr,Atms, ρi, where ρ : Attr → 2Atms, such that A ∈ Attr → 2Atms and ∀a ∈ DOM(A).A(a)∈ρ(a).
Definition 2.11 (s-AVG). An s-AVG is a 5-tuple G = hhAttr,Atms, ρi,AttrPerc, T, P,{S}i, where AttrPerc ⊆Attr,ρ :Attr → 2Atms,S is an s-AVS, and every production rule inPis of the formα→ωi orα0→α1. . . αnwheren≥2,αiis an s-AVS, and (1) ∀a∈DOM(α0)∩AttrPerc.∀1≤i≤n.f ∈
DOM(αi)∧αi(a) =α0(a)
whereα(a)is the value ofain the s-AVSαwith α(a)∈ρ(a).
Intuitively, the AttrPerc features are agreement features whose values percolate up the tree if defined for every level of it.
Example 2.12. Consider the grammar G1 = hh{CAT,PLU,PER},{s,vp,np,v,n, 1,2,3,+,−}, ρi,{PLU,PER},{I,men,John, sleep,sleeps}, P, Si whereρis the specification of appropriate values of attributes:
ρ(CAT) = {s,vp,np,v,n} ρ(PER) = {1,2,3} ρ(PLU) = {+,−}
andP is the set of production rules:
h
CATsi
→ h
CATnpi , h
CATvpi h
CATvpi
→ h
CATvi h
CATnpi
→ h
CATni "
CATn
PER1
#
→ I
"
CATn
PLU+
#
→ men
2 6 4
CATn
PLU-
PER3 3 7
5→ John
"
CATv
PLU+
#
→ sleep
2 6 4
CATv
PLU-
PER3 3 7
5→ sleeps
(1) applies to the subset of attributes{PLU,PER}.
The start symbol isS :h
CATs
i. The grammar gener- ates exactly the sentences:
(2) I sleep.
(3) Men sleep.
(4) John sleeps.
Definition 2.13 (Subsumption). An s-AVS α sub- sumes an s-AVSβ (α ⊑ β) iff∀a.DOM(a).α(a) = β(a).
Definition 2.14 (Derivability). Say G = hhAttr,Atms, ρi,AttrPerc, T, P,{S}i is an s- AVG. If P contains a productionA → ω, then for any φ1, φ2, φ1A′φ2 =⇒3 φ1ω′φ2 if A ⊑ A′ and ω ⊑ ω′. =∗⇒3 is the reflexive, transitive closure of
=⇒3.
Definition 2.15 (s-AVG languages). The language of an s-AVGGis defined as
L(G) = {x∈T∗:∃S′.S′ ⊒S∧S′ =∗⇒3 x}.
2.4 Totally unordered s-AVGs Call totally unordered s-AVGs u-AVGs.
Definition 2.16 (u-AVG). A u-AVG is a 5-tupleG= hhAttr,Atms, ρi,AttrPerc, T, P,{S}i.
Definition 2.17 (Derivability). If A =∗⇒3 ω and ω′ ∈permute(ω), thenA=∗⇒4ω′.
Remark 2.18. (1) means that no Chomsky normal form can be obtained for s-AVG or u-AVG.
3 Bounds on chart size
3.1 Type 2 grammars
Lemma 3.1 (Size of derivation structure). SayD= hV, eiis a derivation structure for ω, GwhereGis a type 2 grammar in Chomsky normal form. It now holds that|V| ≤(3n−1).
Proof. Since Gis in Chomsky normal form, there are only two kinds of production rules: Any deriva- tion ofωof lengthnneedsn−1binary applications, andnunary ones, i.e. of non-branching rules. There arenmany terminals. Consequently, the derivation structure is at most3n−1.
Definition 3.2 (ω-grammar). Say you have a type 2 grammar in Chomsky normal form G = hN, T, P,{S}iand some stringω1. . . ωn. Construct Gω =hNω, Tω, Pω,{1Sn}isuch that
Tω = {ω1, . . . , ωn} and, recursively
(a) (ωi ∈ TωandA → ωi ∈ P) ⇒ (iAi ∈ NωandiAi →ωi ∈Pω)
(b) (iBj,j+1Ck ∈ NωandA → BC) ⇒ (iAk ∈ Nω∧iAk →iBj j+1Ck ∈Pω)
Example 3.3. Consider aabb-grammar of the Chomsky normal form grammar in Example 2.5.
First Tω = {a1, a2, b3, b4}. By (a), the terminal rules are constructed: 1A1 → a1, 2A2 → a2,
3B3→b3, and4B4 →b4. Nonterminal binary rules can now be constructed:
2S3 →2A23B3 2X3 →2A23B3 2T4 →2X34B4 1X4 →1A12T4
1S4 →1A12T4
Naabb = {1A1,2A2,3B3,4B4,2S3,2X3,2T4,
1X4,1S4}.
Our construction of Gωgives us two sets of pos- sible interest,NωandPω. It is easy to see that
|Nω| ≤ |N| ×(n22+n)
where |ω| = n. In our example above this amounts to|Naabb|= 9≤4× 4×25 = 40.
The chart size is bounded by|Pω|+n.
Lemma 3.4 (Chart size). SayGis a type 2 grammar in Chomsky normal andω ∈ T∗. It now holds that
|CG,ω| ≤(|Nω| ×n× |N|2) + (|N| ×n) +n.
Proof. Eachω-nonterminal (|Nω|many) has at most two daughters, and n× |N|nonterminals are non- branching. Since there are at mostnways to split up the span of a branching terminal in two, and at most
|N|2 variable combinations for the two daughters, ((|Nω|)×n× |N|2+ (n× |N|))is clearly an upper bound on|Pω|. In fact, the result is suboptimal, since theiXi-nonterminals count twice.
The number of trees with n leafs is Cn−1 (the Catalan number).
3.2 Totally unordered type 2 grammars
Lemma 3.5 (Size of derivation structure). SayD= hV, eiis a derivation structure forω, GwhereGis a totally unordered type 2 grammar in Chomsky nor- mal form. It now holds that|V| ≤(3n−1).
Proof. Similar to the proof of Lemma 3.1.
Definition 3.6 (ω-grammar). Say you have a to- tally unordered type 2 grammar in Chomsky normal formG=hN, T, P,{S}iand some stringω1. . . ωn. ConstructGω=hNω, Tω, Pω,{S}isuch that
Tω = {ω1, . . . , ωn} and, recursively
(a) (ωi ∈ TωandA → ωi ∈ P) ⇒ (A{i} ∈ Nω andA{i} →ωi∈Pω)
(b) (BΣ, CΣ′ ∈ NωandΣ ∩ Σ′ = ∅andA → BC)⇒ (AΣ∪Σ′ ∈Nω∧AΣ∪Σ′ →BΣCΣ′ ∈ Pω)
Lemma 3.7 (Chart size, upper bound). SayGis a totally unordered type 2 grammar in Chomsky nor- mal and ω ∈ T∗. It now holds that |CG,ω| ≤
|Nω|2×n2× |N|2) + (n× |N|) +n.
Proof. There aren2 ways to split up a sequence in two discontinuous parts.
.
Lemma 3.8 (Chart size, lower bound). Say Gis a totally unordered type 2 grammar in Chomsky nor- mal andω∈T∗. It now holds that|CG,ω| 6∈ O(nk), i.e. chart size is exponential.
Proof. It is easy to see this. You only need to con- sider the upper bound on |Nω| in the totally un- ordered case:
|Nω| ≤ |N| ×2n
3.3 s-AVGs
Lemma 3.9 (Size of derivation structure). SayD= hV, ei is a derivation structure forω, GwhereGis an s-AVG. It now holds that|V| ≤3n−1×(|Attr|+
1).
Definition 3.10 (ω-grammar). Say you have an s-AVG G = hhAttr,Atms, ρi,AttrPerc, T, P,{S}i and some string ω1. . . ωn. Construct Gω = hhAttr,Atmsω, ρi,AttrPerc, Tω, Pω,{1Sn}i such that
Tω = {ω1, . . . , ωn} and, recursively
(a) ([ω]i ∈ Tω andα → [ω]i ∈ P) ⇒ (i[α]i ∈ Nω andi[α]i →[ω]i ∈Pω)
(b) (i[α1]j,j+1[α2]k, . . . ,m−1[αn]m ∈ Nωand[α0] → [α′1][α′2]. . .[α′n] ∈ P and∀1 ≤ i ≤ n.[αi] ⊑ [α′i] ∨ [α′i] ⊑ [αi]) ⇒ (i[α0]m ∈ Nω ∧ i[α0]m →i
[α1]j,j+1[α2]k, . . . ,m−1[αn]m ∈Pω)
I introduce square brackets to enhance readability, i.e. to separate daughter tags from positions. Posi- tions are outside brackets.
We no longer have a set|N|to measure chart size.
The set of possible category structures isAtmsAttr. However, by inspection of our definition of ω, a tighter bound is obtained:
|Nω| ≤ |P| ×(n22+n)
Unfortunately, no such bound can be placed on
|Pω|. The reason is, of course, that productions
i[α0]m → i[α1]. . .[αn]m, and not i[α0]m →
i[α′1]. . .[α′n]mare recorded in Definition 3.10.
Lemma 3.11 (Chart size). Say Gis an s-AVG and ω ∈ T∗. It now holds that|CG,ω| ≤ (|Nω| ×n×
|Atms||Attr|×k) + (|P| ×n) +n, ifGdo not contain m-ary rules such thatm > k.
Proof. Compare the situation to Lemma 3.4.
|Atms||Attr|×k is the number of combinations of daughter categories ink-ary productions.
3.4 Totally unordered s-AVGs
The upper bound on derivation structions in the to- tally unordered case is the same as for s-AVGs. ω- grammars for u-AVG are built analogously to ω- grammars for totally unordered type 2 grammars. It is easy to see that:
|Nω| ≤ |P| ×2n It now holds:
Lemma 3.12 (Chart size). Say Gis an u-AVG and ω ∈ T∗. It now holds that|CG,ω| ≤ (|Nω| ×n×
|Atms||Attr|×k) + (|P| ×n) +n, ifGdo not contain m-ary rules such thatm > k.
In sum,
Theorem 3.13. Totally unordered 2 grammars, s- AVGs and u-AVGs have worst case exponential charts.
This leads us to consider complexity and genera- tive capacity.
3.5 Complexity and generative capacity Consider the universal recognition problem:
Definition 3.14 (Universal recognition). Universal recognition is the decision problem:
INSTANCE: A grammarGand a stringω.
QUESTION: Isωin the language denoted byG?
Lemma 3.15 ((Barton, 1985)). The universal recog- nition problem for totally unordered type 2 gram- mars is NP-complete.
Proof. The vertex cover problem involves finding the smallest setV′of vertices in a graphG=hV, Ei such that every edge has at least one endpoint in the set. Formally,V′ ⊆V :∀{a, b} ∈E:a∈V′∨b∈ V′. The problem is thus an optimization problem, formulated as a decision problem:
INSTANCE: A graphGand a positive integerk.
QUESTION: Is there a vertex cover of size kor less forG?
Say k = 2, V = {a, b, c, d}, E = {(a, c),(b, c),(b, d),(c, d)}. One way to obtain a vertex cover is to go through the edges and underline one endpoint of each edge. If you can do that and only underline two vertex symbols, a vertex cover has been found. Since |V| = 4, this is equivalent to leaving two vertex symbols untouched. Conse- quently, the vertex cover problem for this specific instance is encoded by the totally unordered type 2 grammar, whereδis a bookkeeping dummy symbol:
S → ρ1ρ2ρ3ρ4uuδδδδ ρ1 → a|c
ρ2 → b|c ρ3 → b|d ρ4 → c|d
u → aaaa|bbbb|cccc|dddd δ → a|b|c|d
ρi captures the ith edge in E. The input string ω =aaaabbbbccccdddd. Generally, the first produc- tion has as manyρias there are edges in the graph,
|V|−kmanyu’s and|E|×|V|−|E|−|E|×(|V|−k) manyδ’s, i.e. the length of the string minus the num- ber of edges and the extension of|V| −kmanyu’s.
The ρi productions are simple, u extends into |E|
manya’s orb’s or so on, andδextends into all pos- sible vertices. Since the grammar and input string can be constructed in polynomial time from an un- derlying vertex cover problem hk, V, Ei, universal recognition of UCFG must be at least as hard as solving the vertex cover problem. Since the vertex cover problem is NP-complete (Garey and Johnson, 1979), the universal recognition problem for totally unordered type 2 grammars is accordingly NP-hard.
It is easy to see that it is also in NP. Simply guess a derivation, polynomial in size by Lemma 3.5, and evaluate it in polynomial time.
Lemma 3.16. The universal recognition problem for s-AVGs is NP-complete.
Proof. The 3SAT problem is a variant of the sat- isfiability problem of propositional logic for con- junctions of clauses of three literals, e.g. p∨ p∨ p∧ ¬p∨ ¬p∨ ¬p is not satisfiable in any model.
Its complexity is the same as its older sister’s: It is NP-complete. It is relatively easy to code this problem up in s-AVG. The details are left for the reader. Hint: Introduce agreement features for truth assignments and build ternary phrases that ensure at least one propositional variable in the original prob- lem is true. Since AttrPerc must percolate by (1), you need four rules for each propositional variable (true and false for with and without negation). It follows that the universal recognition problem for s- AVGs is NP-hard. It is easy to see that it is also in NP. Simply guess a derivation, polynomial in size by Lemma 3.9, and evaluate it in polynomial time.
Lemma 3.17. The universal recognition problem for u-AVGs is NP-complete.
Proof. Similar to the proof of Lemma 3.16. Extra features can be used for clause bounds.
Remark 3.18. It is cheap to add linear precedence constraints to totally unordered type 2 grammars and u-AVGs, e.g. to ensure that all verbs precede nouns.
Such constraints can be resolved in time O(n2) on even the most naïve set-up.
If linear precedence constraints are added, it holds that
Lemma 3.19. The totally unordered type 2 lan- guages and the totally unordered simple attribute- value languages both are not included in the type 2 languages.
Proof. Both the totally unordered type 2 languages and the totally unordered simple attribute-value lan- guages include {ambncmdn}. The simplest way to encode it is to let some ruleS →abScd|abcdinter- act with some precedence rule that requires alla’s to precede allb’s, and so on. Similarly, with s-AVSs.
It is just as easy to code up the MIX language, for instance.
4 k-ambiguity
Our strategy to obtain polynomial charts in the to- tally unordered cases is to restrict ambiguity. A rigid lexicon is first imposed. In a rigid lexicon every phonological string is associated with at most one lexical entry.
Remark 4.1. Rigidity is a strong constraint in the absence of inheritance. Inheritance provides an al- ternative to lexical ambiguity, namely underspeci- fication. Such use of inheritance seems necessary for realistic applications ofk-ambiguous grammars.
Rigidity needs only to apply to open class items.
There seems to be some evidence from cognitive neuropsychology that people actually underspecify open class items wrt. morphological features, va- lence and even syntactic category.
The next step is to restrict ambiguity in parsing.
Definition 4.2. A sign is horizontallyk-ambiguous if it only combines with k signs in a sentence. A grammar is horizontallyk-ambiguous if all signs are k-ambiguous. A grammar is verticallyk-ambiguous if signs are combined unambiguously afterksteps.
It is important to remember that our unordered grammars allow signs to combine non-locally. The notion ofk-ambiguity can be illustrated by an exam- ple from Icelandic:
Example 4.3. Icelandic has nominative objects.
Consider, for instance:
(5) Hún she
spurði asked
hvort whether
sá the.NOM
grunaði suspected.NOM
væri
was.3SG.SUBJ
örugglega surely
þú.
you.SG.NOM
’She asked whether the suspect surely was you.’
In addition, both SVO and OVS constructions oc- cur. So in many cases, a verb that seeks to combine with an object has more than one candidate for doing so, even in sentences with only three constituents:
NP.NOM V. NP.NOM
The V constituent is said to be horizontally 2- ambiguous in this case.
For simplicity, the notion of the order of an s-AVS is introduced:
Definition 4.4. An s-AVSαis said to be of orderl iff|DOM(α)|= l. If all s-AVSs in a grammarGare of orderl,Gis itself said to be of order 1.
Lemma 4.5. Type 2 grammars are equivalent to s- AVGs of order 1. Totally unordered type 2 languages are equivalent to u-AVGs of order 1.
Proof. Trivial.
Say s-AVSs are of order 1, and vertical ambiguity 1(i.e. horizontal ambiguityk). We then have:
|CG,ω| ≤ (n22−n+
i<n
X
1<i
(kn(n−i))) +n
First all initial combinations n22−n are checked.
At this point, there can be at mostkncandidate mod- els. For each candidate model, the next set of com- binations is checked. Since vertical ambiguity is1, the set of candidate models remains at mostkn.
If we fix vertical ambiguity to k (i.e. horizontal ambiguityn):
|CG,ω| ≤ (n22−n +
i<n
X
1<i
(nk(n−i))) +n
which is inO(nk+2). Since the order of s-AVSs is bound by|Attr|, it holds that:
Theorem 4.6. k-ambiguous totally unordered 2 grammars, k-ambiguous s-AVGs andk-ambiguous u-AVGs have polynomial charts.
Proof. See above. The result for s-AVGs is sub- sumed by the result for u-AVGs.
Remark 4.7. For unordered type 2 grammars, and possibly for totally unordered ones too, it is an alter- native to say that all totally unordered productions in a chart have a yield of at mostk. This gives a bound on chart size:
0≤i
X
i<k
(|N| ×(n−i)×
0≤j
X
j<(k−i)
(|N|k−j)×(k−i)×(n−j))
+
k<i
X
i<n
(|N|3×(n−i))
This fragment no longer generates the MIX lan- guage. Such a constraint is obviously not enough for u-AVG, since s-AVG is NP-complete. A third possibility is to restrict the arity of productions.
5 Conclusions and related work
In last year’s conference, Søgaard and Haugereid (2006) presented, in a rather infor- mal fashion, a restrictive typed attribute-value structure grammar formalismTf for free word order phenomena, equipped with a polynomial parsing algorithm. In Tf, horizontal k = 1. The purpose of their paper was mostly philosophical, i.e. in favor of underspecification rather than ambiguity, but many details were left unclear. In a sense, this paper provides a formal basis for some of the claims made in that paper. In particular, types are easily added to s-AVG and u-AVG, and more flexible attribute-value structures can be employed (as long as they are at most polynomial in the size of strings).
UnlikeTf,k-ambiguous grammars also admit fixed ambiguity.
Other researchers have tried to single out tractable attribute-value grammars:
Seki et al. (1993) operate in the context of LFG.
For a start, they restrict the expressive power of LFG by restricting the syntax of LFG-style func- tional schemas to:
(↑attr=val)or(↑attr=↓)
Call this fragment non-deterministic copying LFG (nc-LFG). They then proceed to define two tractable fragments of nc-LFG:
Definition 5.1. An nc-LFG is called a dc-LFG (de- terministic . . . ) if each pair of rulesr1 : A → α1
and r2 : A → α2 whose left-hand sides are the same is inconsistent in the sense that there exists no f-structure that locally satisfies both of the functional schemata ofr1 andr2.
Definition 5.2. An nc-LFG is called a fc-LFG (fi- nite . . . ) if it contains only a finite number of so- called “subphrase nonterminal” (SPN) multisets, i.e.
a multiset of nonterminals N such that there ex- ists consistent productions A1 → α1. . . An. . . αn and an attribute attr such that N = {αi ∈ {α1. . . αn}|(↑attr=↓)is the FS ofαi}.
A nice example of an nc-LFG that is not an fc- LFG is mentioned in (Seki et al., 1993):
Example 5.3. LetGbe an nc-LFG whereN ={S}, T = {a}, Lbls = {log}, e the only value, and productions are:
S → S S
(↑log=↓) (↑log=↓)
S → a
(↑log=e)
Gis not an fc-LFG, since the SPN multisets inG include
{{S}},{{S, S}},{{S, S, S, S}}, . . .. Both fragments are tractable, and the weak gen- erative capacity of dc-LFG is equivalent to that of finite-state translation systems, while the weak gen- erative capacity of fc-LFG is equivalent to that of linear indexed grammars. It follows that fc-LFG is also equivalent to one-reentrant attribute-value grammar (Feinstein and Wintner, 2006).
Keller and Weir (1995) go beyond linear indexed grammars on their way toward attribute-value gram- mar. The first step on this path is to replace the stacks of indeces in linear indexed grammars with trees. Tractability is ensured by the requirement that subtrees of any mother that are passed to daughters that share subtrees with one another must appear as siblings in the mother’s tree. The following such grammar generates{anbncn}:
S1[σ0] → A[x]S2[σ(x, x)]
S2[σ(x, y)] → B[x]S3[y]
S3[x] → C[x]
A[σ2(x)] → aA[x]
B[σ2(x)] → bB[x]
C[σ2(x)] → cC[x]
A[σ1] → a B[σ1] → b C[σ1] → c
In a sense, this is much like s-AVG, except that reentrancies replace (1) and roots cannot be reen- tered. Keller and Weir argue this is no problem if
the entire structure is seen as the derivational output, rather than just the AVS of the mother. In addition, reentrancy is interpreted intensionally in their set- up, rather than extensionally. This is similar to ours.
Both formalisms are stronger than k-ambiguous u-AVG in some respects. This is easy to see. Both nc-LFG and Keller and Weir’s richer fragment of attribute-value grammar are superfinite, i.e. they generate all finite languages. k-ambiguous u-AVG doesn’t. It holds that:
Lemma 5.4. Thek-ambiguous u-AVG languages do not include (all of) the regular languages.
The proof is omitted, but consider the simpler proof of:
Lemma 5.5. The1-ambiguous u-AVG languages do not include (all of) the regular languages.
Proof. Consider the language
a{b|. . .|n} ∪ p{b|. . .|n} ∪ {b|. . .|n}
but notj but noti
in which ab, ai, pj, b are strings, while aj, pi, bb are not. This language is regular, but cannot be gen- erated by a1-ambiguous u-AVG.
It should be relatively easy to see how this gener- alizes tok-ambiguous u-AVG.
In sum, it was shown that the exponential worst case complexity of totally unordered charts is dram- maticaly reduced by the k-ambiguity constraint.
In particular k-ambiguous charts are in O(nk+2).
Since subsumption is linear time solvable, the recog- nition problem for k-ambiguous u-AVGs is also solvable in polynomial time. Efficient algorithms and their complexity are the topic of future pub- lications. k-ambiguous u-AVG differs in signif- icant ways from other polynomial time attribute- value grammars. In particular,k-ambiguous u-AVG was designed for analyses of discontinuous con- stituency. It provides the formal machinery needed for “crossed branches” analyses. In addition, k- ambiguous u-AVG is not superfinite. It is conjec- tured – also by one of the reviewers – that this has interesting consequences for learnability.
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